Problem 11 - Beginning and Intermediate Algebra

Question

Simplify. $$5^{8} \cdot 5^{-6}$$

Step-by-Step Solution

Answer

Using the product rule of exponents, the simplified expression is: \(5^{8} \cdot 5^{-6} = 5^{8-6} = 5^2 = 25\).

Step 1: Write down the expression

The given expression is: \(5^{8} \cdot 5^{-6}\).

Step 2: Apply product rule of exponents

Since both terms have the same base (5) and are being multiplied, we can use the product rule of exponents to simplify the expression. The rule states: $$a^m \cdot a^n = a^{m+n}$$ So, in our case: $$5^8 \cdot 5^{-6} = 5^{8+(-6)}$$

Step 3: Add the exponents

Now, we just need to add the exponents: $$5^{8+(-6)} = 5^{8-6}$$ $$5^{8-6} = 5^2$$

Step 4: Calculate the result

Now, we can calculate the result: $$5^2 = 25$$ So, the simplified expression is 25.

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental part of algebra that involves reducing complex algebraic expressions to their simplest form. When working with exponents, this often means applying specific rules to combine and reduce terms. Understanding how to simplify expressions is essential for solving more complex problems and making algebra easier to handle.

For instance, if you are given an expression with the same base, you can consolidate it by manipulating the exponents. Simplification can involve various techniques, such as:

• Combining like terms
• Using the product and quotient rules for exponents
• Applying the properties of exponents, such as power of a power

In the example of simplifying \(5^8 \cdot 5^{-6}\), we focus on one main rule of exponents to make the expression simpler.

Product Rule

The product rule is a powerful tool when simplifying expressions that involve exponents. It states that whenever you multiply terms with the same base, you can add their exponents together. This rule is expressed as:

\[a^m \cdot a^n = a^{m+n}\]

Here, \(a\) represents the base, and \(m\) and \(n\) are the exponents. By understanding the product rule, you can effectively reduce multiplication of powers to a single term with one exponent, making calculations simpler.

In our given exercise, \(5^8 \cdot 5^{-6}\), both numbers have the base of 5. Using the product rule allows us to combine the exponents: \(8\) and \(-6\). The expression then transforms into \(5^{8 + (-6)}\) or \(5^2\), significantly simplifying the original expression.

Base and Exponent Concepts

In algebra, understanding the roles of the base and exponent is crucial to working with powers. The base is the number that is being multiplied by itself, and the exponent indicates how many times it is multiplied.

For example, in \(5^8\), 5 is the base, and 8 is the exponent, meaning 5 is multiplied by itself 8 times. When dealing with different operations, especially in expressions like \(5^8 \cdot 5^{-6}\), it's important to remember:

• Same base allows exponent rules to be applied.
• Negative exponents imply a reciprocal operation.
• Zero as an exponent results in 1.

Understanding these concepts helps in applying the correct rules to simplify expressions and handle algebraic operations more efficiently.

Algebraic Operations

Algebraic operations involve performing mathematical activities, such as addition, subtraction, multiplication, and division, on expressions. When dealing with exponents, these operations require a good grasp of exponent rules to execute them correctly.

In our example, multiplying powers of 5, the operation is initially straightforward, but because the exponents include a negative number, employing algebraic rules becomes necessary. Here's how algebraic operations on exponents simplify the task:

• Convert multiplication of same bases to addition of exponents.
• Simplify negative exponents using reciprocal relationships.
• Cancel terms using subtraction where applicable.

Thus, algebraic operations in expressions like \(5^8 \cdot 5^{-6}\) extend beyond basic multiplication; they incorporate exponent rules to yield a simple, elegant solution: \(5^2 = 25\). These skills make tackling more complex algebra problems much easier.